cm cm
V^ff
where E
Icob lb lcol leff !
modulus of elasticity of the concrete (see 3.1.2.5.2) moment of inertia (gross section) of the column or beam respectively height of the column measured between centres of restraint effective span of the beam factor taking into account the conditions of restraint of the beam at the opposite end: ! = 1.0 opposite end elastically or rigidly restrained ! = 0.5 opposite end free to rotate ! = 0 for a cantilever beam.
(2) Isolated columns are considered slender if the slenderness ratio of the column considered exceeds 25 or 15Vvu, whichever is the greater, where:
vu the longitudinal force coefficient for the element:
2 slenderness ratio: 2 = lo/i lo effective height or length of the vertical element. lo is generally derived using the elastic buckling theory. For frame structures, the column to which lo refers needs to be defined carefully. i radius of gyration.
P(1) Allowance shall be made in the design for the uncertainties associated with the prediction of second order effects and, in particular, dimensional inaccuracies and uncertainties in the position and line of action of the axial loads. In the absence of other adequate provisions, this may be achieved by the use of equivalent geometrical imperfections.
(2) For frame structures an inclination v of the complete structure (bracing elements and braced sub-assembly) from the vertical is defined in 2.5.1.3.
(3) For isolated elements, the equivalent geometrical imperfections may be introduced by increasing the eccentricity of the longitudinal force by an additional eccentricity ea, acting in the most unfavourable direction:
where lo denotes the effective length of the isolated element (see 4.3.5.3.5) v inclination from the vertical calculated using Equation (2.10).
4.3.5.5 Specific data for different types of structure
4.3.5.5.1 Non-sway frames
P(1) Individual non-sway compression members shall be considered to be isolated elements and be designed accordingly.
P(2) Bracing elements or, in non-sway frames without bracing elements, the individual compression members shall be designed for the relevant horizontal forces and vertical loads taking account of the equivalent geometrical imperfections defined in 2.5.1.3 and 4.3.5.4 respectively.
(3) For individual compression members, the design rules for isolated columns (see 4.3.5.5.3) apply. The effective length lo may generally be determined according to 4.3.5.3.5.
4.3.5.5.2 Sway frames
(1) Information on sway frames is given in Appendix 3.
P(1) Second order effects including geometrical imperfections and deformations due to creep, if they affect the structural stability significantly, shall be taken into account in the design of slender isolated compression members.
(2) Isolated columns in non-sway structures need not be checked for second order effects if the slenderness ratio 2 is less than or equal to the value give in Equation (4.62) even though the column may be classified as slender by 4.3.5.3.5.
eo1 and eo2 are the eccentricities of the axial load at the ends of the member and it is assumed that |eo1| r leo2 |
In this case, the column ends should be designed for at least the conditions given by Equations (4.63) and (4.64).
where NRd is the design resisting axial force in compression and MRd is the design resisting moment.
Equation (4.62) should be used only if the column is not subjected to transverse loads between its two ends.
The criterion defined by Equation (4.62) is shown graphically in Figure 4.28. For the design of columns see 4.3.5.6.
P(3) For columns bent dominantly about one principal axis, the possibility of failure due to second order effects about the second principal axis should be checked.
(4) For this check the initial eccentricity eo in the direction of the second principal axis should be taken as zero and the second order effects should be calculated by using the slenderness ratio 2 related to this axis. The additional eccentricity defined in 4.3.5.4(3) and, where relevant, creep deformations should be taken into account.
a) Structural system b) Idealisation of the column considered c) Critical slenderness ratio 2crit a) Structural system b) Idealisation of the column considered c) Critical slenderness ratio 2crit
Figure 4.28 — Slenderness limits for isolated members with rigidly or elastically restrained ends in non-sway structures
P(5) Principles (1) and (3) above also apply to compression members subjected to bi-axial bending in which the effects of torsion are negligible.
(6) If the first order eccentricity eo of the axial force in the direction of the first principal axis exceeds eo > 0.2 h, the check in the direction of the second principal axis should be based on the reduced section depth h' as defined in 4.3.5.6.4(3).
4.3.5.6 Simplified design methods for isolated columns
(1) For buildings, a design method may be used which assumes the compression members to be isolated and adopts a simplified shape for the deformed axis of the column. The additional eccentricity is then calculated as a function of the slenderness.
(1) The total eccentricity attributed to columns of constant cross-section (concrete and steel, ignoring laps) in the most heavily stressed section (critical section) is given by:
a) First order eccentricities, equal at both ends [Figure 4.29(a)]:
where eo first order eccentricity:
eo = MSd1/NSd Msd1 first order applied moment Nsd applied longitudinal force ea additional eccentricity according to Equation (4.61)
e2 second order eccentricity, using the approximate methods given in 4.3.5.6.3 below, including any effects due to creep.
b) Where the first order eccentricites are different at both ends [Figure 4.29(b), Figure 4.29(c)]
For columns of constant cross-section (concrete and steel ignoring laps) subjected to first order moments varying linearly along their length and having eccentricities at their ends which differ in value and/or sign, an equivalent eccentricity ee should be used in Equation (4.65), instead of eo, for the critical section.
The equivalent eccentricity ee can be taken as the higher of the following values:
where eo l and eo 2 denote the first order eccentricities at the two ends, and |eo2l T leol | [Figure 4.29(b) and Figure 4.29(c)]
4.3.5.6.3 Model column method a) Scope and definition
1) The design method which is described below relates to members where 2 < | 140 | and with rectangular or circular cross-sections and in which the first order eccentricity satisfies the condition eo T 0.1 h (h: depth of the cross section measured in the plane under consideration). For other shapes of cross-section and for the eccentricities eo < 0.1 h, other appropriate approximations may be used.
2) A "model column" is a cantilever column which is:
— fixed at the base and free at the top (Figure 4.30):
— bent in simple curvature under loads and moments which give the maximum moments at the base.
The maximum deflection, which equals the second order eccentricity, e2 of such a column may be assumed to be:
where lo is the effective length of the column and l/r is defined in 3) below
3) The stability is analysed on the basis of the curvature l/r in the critical section at the base. This curvature is derived from the equilibrium of the internal and external forces.
b) Transformation of second order analysis to cross section design
5) In cases where great accuracy is not required, the curvature 1/r in Equation (4.69) may be derived from
where eyd is the design yield strain of steel reinforcement = fyd/Es d is the effective depth of the cross-section in the expected direction of stability failure.
6) The coefficient K2 in Equation (4.72) takes account of the decrease of the curvature 1/r with increasing axial force and is defined by
where
Nud is the design ultimate capacity of the section subjected to axial load only. It may be taken as Nud = a.fcd.Ac + fyd.As. [for a see 4.2.1.3.3(11)]
Nbal is the axial load which, when applied to a section, maximises its ultimate moment capacity. For symmetrically reinforced rectangular sections, it may be taken as 0.4.fcd Ac.
It will always be conservative to assume that K2 = 1 4.3.5.6.4 Compression members with bi-axial eccentricities
(1) For members with rectangular cross sections, separate checks in the two principal planes y and z (see Figure 4.31) are permissible if the ratio of the corresponding eccentricities ey/b and ez/h satisfy one of the following conditions:
(i.e. that the point of application of NSd is located in the hatched area in Figure 4.31). The eccentricities ey and ez are the first order eccentricities in directions of the section dimensions b and h respectively. They need not include ea as defined in Equation (4.61). A refined analysis will be required if the stated conditions are not met.
(2) For the two separate checks 4.3.5.3.5 (effective heights and slenderness limits), 4.3.5.5.3
and 4.3.5.6.2 — 4.3.5.6.3 apply analogously where the slenderness limits given in 4.3.5.3 are exceeded. The geometrical imperfections defined in 4.3.5.4 should, however, be considered in the two planes.
(3) Where ez > 0.2 h (see Figure 4.32), separate checks are permissible only if the check for bending about the minor axis of the cross section (z in Figure 4.31) is based on the reduced width h' as shown in Figure 4.32. The value h' may be determined on the assumption of a linear stress distribution e.g. from:
where:
NSd longitudinal force, negative sign in compression. Zc modulus of gross section.
eaz additional eccentricity according to Equation (4.61) in the z direction.
(4) If the criterion in (1) above is not met, a refined analysis is necessary.
4.3.5.7 Lateral buckling of slender beams
P(1) Where the safety of beams against lateral buckling is in doubt, it shall be checked by an appropriate method.
(2) The safety against lateral buckling of reinforced and prestressed concrete beams may be assumed to be adequate if the requirements in Equation (4.77) are satisfied. Otherwise a more detailed analysis should be carried out.
b width of the compression flange h total depth of the beam lot length of the compression flange measured between lateral supports.
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